Archimedes was born in the city of Syracuse on the island of Sicily in 287 BC. As a youth in Syracuse Archimedes had developed a natural curiosity for problem solving. Once Archimedes felt like he learned as much as he could in Syracuse, he went to Egypt to study in Alexandria. Archimedes was undoubtedly influenced by the works of Euclid (Dunham, "Journey Through Genius"). After Archimedes was done studying at Alexandria, he went back to Syracuse and began to problem solve for King Hiero II. One of the problems that Archimedes worked on for King Hiero is sometimes referred to as " The Puzzle of King Hiero's Crown". The King had ordered a crown to be made for him out of solid gold, which he supplied to the blacksmith. Once The king received the crown, he had suspicion that the crown was not pure gold, and that the blacksmith kept most of the gold for himself, and made the crown out of a little bit of the gold and mostly silver. Archimedes was tasked with determining if the crown was solid gold without damaging the crown in the process. Archimedes actually found the solution to this problem while taking a bath. He noticed that the full bath overflowed when he lowered himself into it, and realized that he could measure the crown's volume by the amount of water it displaced. He knew that since he could measure the crown's volume, all he had to do was discover its weight in order to calculate its density and hence its purity.
In 1906 unknown mathematical works of Archimedes was discovered, and is referred to as the "Archimedes Palimpsest". Archimedes is considered to have been one of the greatest mathematicians of antiquity. Archimedes produced formulas to calculate the areas of regular shapes, using a revolutionary method of capturing new shapes by using shapes he already understood ( Mastin, "The Story of Mathematics"). This method of using already known shapes in order to approximate the area of a circle is referred to as "Archimede's Method". Archimedes methodology for calculating pi was so powerful that in the 16th century Ludoph can Ceulun used a polygon with an extraordinary 4,611,686,018,427,387,904 sides to arrive at pi correct to 35 digits! Archimedes most impressive use of "Archimede's method" was his proof known as the Quadrature of the Parabola, which states that the area of a parabolic segment is 4/3 that of a certain inscribed triangle. Many also believe that Archimedes had the most prescient view of the concept of infinity of all Greek mathematicians.
Brahmagupta was (possibly) born in Ujjain, India, in 598. Brahmagupta wrote very important works on both mathematics and astronomy, and became the head of the astronomical observatory at Ujjain in central India. Most of his works are composed in elliptic verse which was a common practice in Indian mathematics at the time (Luke, "Life of Brahmagupta"). Brahmagupta's text "Brahmasphutasidddhanta" is probably the earliest known text to treat zero as a number (class handout, Luke, "Life of Brahmagupta"). Brahmagupta also had alot of work on arithmetic, explaining how to find the cube-root of an integer and gave rules facilitating the computation of the squares and square roots (Mason, "Indian Mathematics"). He also did alot of work with fractions. Brahmagupta gave the sum of the squares of the first n natural numbers and the sum of the cubes of the first n natural numbers. Brahmagupta established the basic mathematical rules for dealing with zero (1+0=1; 1-0=1; and 1 x 0 = 0). Brahmagupta even wrote down abstract concepts using the initials of the names of colors to represent unknowns in his equations, which is one earliest signs of what we now know as algebra. Brahmagupta had many major discoveries; he was the first to discover the formula for solving quadratic equations. He wrote that the length of a year was 365 days 6 hours 12 minutes 9 seconds, and many other properties dealing with both positive and negative numbers.
Omar Khayam was a Persian mathematician, philosopher, poet and astronomer born in 1048 in Nishpur (modern day Iran). Omar started his career by teaching algebra and geometry. One of Khayam's most famous works include a treatise called "Treatise on Demonstration of Problems of Algebra". He also laid the foundation of Pascal's triangles (even though Pascal's Triangle was discussed by al-Karaji, before Omar's time) with his work on triangular array of binomial coefficients. Another major work was titled "Explanations of the Difficulties in the Postulates of Euclid" It is believe that Omar Khayyam was trying to prove Euclid's parallel postulate when ended up proving the properties of figures in the non-euclidean geometry.
All three men, Archimedes, Brahmagrupta. and Omar Khayam, were revolutionaries of their time, and helped pave the way to modern day mathematics.
Nice research on these fellows. The Palimpsest is fascinating as well as well-named. Good mathy tidbits, as well. To be an exemplar, you could strengthen the consolidation: what was it about these fellows that supported modern math? It doesn't seem like you mean the bits of math, but maybe you do?
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The first example that comes to my mind of how Archimedes has impacted modern math was his invention of "Archimedes screw", used for lifting water to higher levels, which is still used today in some areas, which is pretty impressive considering how long ago it was that he invented it. Also Archimedes' exhaustive proof methods are a method of proof that is still used today.
DeleteFor Brahmagupta, his applications of mathematics to astronomy had an immense effect on many other mathematicians and astronomists, including al-Khawrizmi, the "father of algebra". He was basically responsible for creating the basic rules of arithmetic, specifically multiplication of positive, negative, and even, zero values. These tactics and rules are still used in modern math.
And Omar Khayyan's work of having a hand in creating a general binomial theorem to extract roots still plays an impact in modern math. For all three of these guys, I think it was their approach of acquiring, and requiring, extensive proofs to mathematical claims, and tended to question and challenge propositions, and sometimes ended up finding proofs for other propositions instead. I think this approach to mathematics never died with these guys, and is still strongly supported in today's modern math. My opinion.